ON SPATIAL ENTROPY OF MULTI-DIMENSIONAL SYMBOLIC DYNAMICAL SYSTEMS

The commonly used topological entropy h(top) (U) of the multidimensional shift space U is the rectangular spatial entropy h(r) (U) which is the limit of growth rate of admissible local patterns on finite rectangular sub-lattices which expands to whole space Z(d), d >= 2. This work studies spatial entropy h(Omega)(U) of shift space U on general expanding system Omega = {Omega(n)}(n=1)(infinity) where Omega(n) is increasing fi nite sublattices and expands to Z(d). Omega is called genuinely d-dimensional if Omega(n) contains no lower-dimensional part whose size is comparable to that of its d-dimensional part. We show that h(r)(U) is the supremum of h(Omega)(U) for all genuinely d-dimensional Omega. Furthermore, when Omega is genuinely d-dimensional and satis fi es certain conditions, then h(Omega) (U) = h(r)(U). On the contrary, when Omega(n) contains a lower-dimensional part which is comparable to its d-dimensional part, then h r (U) < h(Omega)(U) for some U. Therefore, h(r)(U) is appropriate to be the d-dimensional spatial entropy.